Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \neq 0$. $n = \dfrac{x^2 + 10x}{x^3 + 15x^2 + 50x} \div \dfrac{3x - 18}{4x^2 - 32x + 48} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $n = \dfrac{x^2 + 10x}{x^3 + 15x^2 + 50x} \times \dfrac{4x^2 - 32x + 48}{3x - 18} $ First factor out any common factors. $n = \dfrac{x(x + 10)}{x(x^2 + 15x + 50)} \times \dfrac{4(x^2 - 8x + 12)}{3(x - 6)} $ Then factor the quadratic expressions. $n = \dfrac {x(x + 10)} {x(x + 10)(x + 5)} \times \dfrac {4(x - 6)(x - 2)} {3(x - 6)} $ Then multiply the two numerators and multiply the two denominators. $n = \dfrac {x(x + 10) \times 4(x - 6)(x - 2) } { x(x + 10)(x + 5) \times 3(x - 6)} $ $n = \dfrac {4x(x - 6)(x - 2)(x + 10)} {3x(x + 10)(x + 5)(x - 6)} $ Notice that $(x + 10)$ and $(x - 6)$ appear in both the numerator and denominator so we can cancel them. $n = \dfrac {4x(x - 6)(x - 2)\cancel{(x + 10)}} {3x\cancel{(x + 10)}(x + 5)(x - 6)} $ We are dividing by $x + 10$ , so $x + 10 \neq 0$ Therefore, $x \neq -10$ $n = \dfrac {4x\cancel{(x - 6)}(x - 2)\cancel{(x + 10)}} {3x\cancel{(x + 10)}(x + 5)\cancel{(x - 6)}} $ We are dividing by $x - 6$ , so $x - 6 \neq 0$ Therefore, $x \neq 6$ $n = \dfrac {4x(x - 2)} {3x(x + 5)} $ $ n = \dfrac{4(x - 2)}{3(x + 5)}; x \neq -10; x \neq 6 $